+467 A four-digit hexadecimal integer is written on a napkin such that the units digit is illegible. The first three digits are $7$, $D$, and $3$. If the integer is a multiple of $25_{10}$, what is the units digit?
+1786 Let's work on an equation for this problem. Let's let the missing units digit number be x.
Since we converting from base 16 (hex) to base 10,
We have the conversion process
\((7D3x)_{16} = (7 × 16^3) + (13 × 16^2) + (3 × 16^1) + (x × 16^0)\)
Now, calculating the first 3 terms of the numbers, we find that
\((7 × 16^3) + (13 × 16^2) + (3 × 16^1)=32048\)
So, this means that \(32048 + x\) has to be divisble by 25.
The closest number that satsifies this is \(32050\), which means that x is 2.
The next closest number is 32075, which would mean x is 27, which is our of range for base 10.
Thus, 2 is our final answer.
Thanks! :)
